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Butcher series are combinatorial devices used in the study of numericalmethods for differential equations evolving on vector spaces. More precisely,they are formal series developments of differential operators indexed overrooted trees, and can be used to represent a large class of numerical methods.The theory of backward error analysis for differential equations has aparticularly nice description when applied to methods represented by Butcherseries. For the study of differential equations evolving on more generalmanifolds, a generalization of Butcher series has been introduced, calledLie--Butcher series. This paper presents the theory of backward error analysisfor methods based on Lie--Butcher series.

Backward error analysis and the substitution law for Lie group integrators